Surreal exponentiation: It gets worse
Feb. 17th, 2012 06:02 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo in the previous entry I asked the question, what the hell is natural exponentation, and does it agree with surreal exponentiation? This of course assumes that there is such a thing as "surreal exponentiation". After all, the surreals are always described as a place where crazy expressions like (ω+3)√π make sense, so everyone has to agree what surreal exponentiation means, right?
Well, now I'm not so certain. There's a definition due to Kruskal and a definition due to Gonshor and I have no idea if these are actually the same. They both have the required properties. (Note these are definitions of exp, not of ^, but since both have exp being onto the positive surreals one can therefore easily turn this into a definition of ^.) There may also be one due to Alling? Oy... though that one at least seems to be similar to Gonshor's...
Then Wikipedia -- not citing any source -- lists a recursive definition of 2x, which, I suppose, could be generalized to ax by just replacing "2" by "a". Does this agree with the others? Again, no idea.
Let's note here something that doesn't fit in -- the function ωx, commonly used in studying surreals, is *not* part of what I'm looking for. It agrees with the ordinary ωx when x is an ordinal, and is not supposed to fit in with any more general notion of exponentiation; the above definitions of exponentiation actually do not agree with it. Wikipedia's definition of 2x seems rather similar to the recursive definition of ωx so maybe it agrees with *that* and ordinary ordinal exponentiation instead? :-/ But that would be a surprise -- though there have been some extensions of ordinary ordinal operations to surreals, but these are not actually well-defined on all surreals.
And then on top of that we still have the question of whether any of these agree with "natural exponentiation" -- whatever the hell that is!
Guess I can try reading the original German for an answer to that latter question, anyway. Well, once the people at the library get SpringerLink working here again...
-Harry
Well, now I'm not so certain. There's a definition due to Kruskal and a definition due to Gonshor and I have no idea if these are actually the same. They both have the required properties. (Note these are definitions of exp, not of ^, but since both have exp being onto the positive surreals one can therefore easily turn this into a definition of ^.) There may also be one due to Alling? Oy... though that one at least seems to be similar to Gonshor's...
Then Wikipedia -- not citing any source -- lists a recursive definition of 2x, which, I suppose, could be generalized to ax by just replacing "2" by "a". Does this agree with the others? Again, no idea.
Let's note here something that doesn't fit in -- the function ωx, commonly used in studying surreals, is *not* part of what I'm looking for. It agrees with the ordinary ωx when x is an ordinal, and is not supposed to fit in with any more general notion of exponentiation; the above definitions of exponentiation actually do not agree with it. Wikipedia's definition of 2x seems rather similar to the recursive definition of ωx so maybe it agrees with *that* and ordinary ordinal exponentiation instead? :-/ But that would be a surprise -- though there have been some extensions of ordinary ordinal operations to surreals, but these are not actually well-defined on all surreals.
And then on top of that we still have the question of whether any of these agree with "natural exponentiation" -- whatever the hell that is!
Guess I can try reading the original German for an answer to that latter question, anyway. Well, once the people at the library get SpringerLink working here again...
-Harry
no subject
Date: 2012-02-18 08:42 am (UTC)exponentiation :-/
no subject
Date: 2012-02-18 08:49 am (UTC)The introduction to Gonshor's book certainly make it sound like his exponentiation is the same as Kruskal's, but he never exactly states it explicitly...
In any case I think Wikipedia may have sent me on something of a wild goose chase... more on that in a moment.
no subject
Date: 2012-02-18 09:07 am (UTC)We have free DigitZeitschriften:
http://www.digizeitschriften.de/main/zeitschriften/
To read Polish papers we have Kolekcja Matematyczna:
http://matwbn.icm.edu.pl/
and for the French NUMDAM
http://www.numdam.org/spip.php?rubrique4&lang=en